Question

The diameters of bolts produced in a machine shop are normally distributed with a mean of 5.34 millimeters and a standard deviation of 0.03 millimeters. Find the two diameters that separate the top 8% and the bottom 8%. These diameters could serve as limits used to identify which bolts should be rejected. Round your answer to the nearest hundredth, if necessary.

Answer 1

Using the normal distribution, we have that:

• The diameter that separates the top 8% is of 5.38 mm.

• The diameter that separates the bottom 8% is of 5.30 mm.

Normal Probability Distribution

The z-score of a measure X of a normally distributed variable with mean $\mu$ and standard deviation $\sigma$ is given by:

$Z = \frac{X - \mu}{\sigma}$

• The z-score measures how many standard deviations the measure is above or below the mean.

• Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.

The mean and the standard deviation for this problem are given, respectively, by:

$\mu = 5.34, \sigma = 0.03$

The 8th percentile separates the bottom 8%, that is, X when Z = -1.405, hence:

$Z = \frac{X - \mu}{\sigma}$

$-1.405 = \frac{X - 5.34}{0.03}$

X - 5.34 = -1.405 x 0.03

X = 5.30.

The 92th percentile separates the top 8%, that is, X when Z = 1.405, hence:

$Z = \frac{X - \mu}{\sigma}$

$1.405 = \frac{X - 5.34}{0.03}$

X - 5.34 = 1.405 x 0.03

X = 5.38.

More can be learned about the normal distribution at https://brainly.com/question/4079902

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